My favorie Neil Armstrong anecdote has very little to do with Neil Armstrong.
It's the scene in Apollo 13 when it looks like the crew isn't going to make it back, and Armstrong and Aldrin go to the Lovells' house to provide moral support. When Lovell's wife introduces Armstrong and Aldrin to Lovell's mother, she asks "Are you boys in the space program too?"
"That's one small step for a man, one giant leap for mankind."
--Boink